- #1
nomadreid
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If there exists a real-valued measurable cardinal, then there is a countably additive extension of Lebesgue measure to all sets of real numbers. This would include then the Vitali sets, which are an example of sets that are not Lebesgue measurable for weaker assumptions than the existence of a real-valued measureable cardinal. However, after going over the proof that a Vitali set is not measurable, for example in Wikipedia's "Vitali set", I do not see where the proof would fail under the assumption of a real-valued measureable, i.e., assuming that there exists a cardinal κ so that there is an atomless κ-additive measure on the power set of κ. I presume I am missing something breathtakingly obvious. Could someone point this out to me?